= Rolling-time factor on a straight incline
{title2=$t_{\rm roll}/t_{\rm slide}=\sqrt{1+I/(MR^2)}$}
For a rigid body rolling from rest down a straight incline with fixed rolling radius $R$ and axial <moment of inertia> $I$, <translation> and <rotation> give $(M+I/R^2)\ddot s=Mg\sin\theta$. A smooth slide has $\ddot s=g\sin\theta$. Travel over the same distance from rest therefore gives the displayed time ratio. <Static friction> must suffice for <rolling without slipping>; its contact point is at rest, so this ideal friction does no work. A uniform solid cylinder has $I=MR^2/2$ and ratio $\sqrt{3/2}$.
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